The chromatic index of graphs with large maximum degree, where the number of vertices of maximum degree is relatively small.

Chetwynd, Amanda G. and Hilton, A. J. W. (1990) The chromatic index of graphs with large maximum degree, where the number of vertices of maximum degree is relatively small. Journal of Combinatorial Theory, Series B, 48 (1). ISSN 0095-8956

Abstract

By Vizing's theorem, the chromatic index χ′(G) of a simple graph G satisfies Δ(G) ≤ χ′(G) ≤ Δ(G) + 1; if χ′(G) = Δ(G), then G is Class 1, and if χ′(G) = Δ(G) + 1, then G is Class 2. We describe the structure of Class 2 graphs satisfying the inequality , where r is the number of vertices of maximum degree. A graph G is critical if G is Class 2 and χ′(H) < χ′(G) for all proper subgraphs H of G. We also describe the structure of critical graphs satisfying the inequality above. We also deduce, as a corollary, an earlier result of ours that a regular graph G of even order satisfying is Class 1.